The symmetric Hessian for an implicit surface defined from a field variable $c(x,y,z)$ in Cartesian space is,
$$ \nabla^2 c = \begin{bmatrix} c_{xx} & c_{xy} & c_{xz} \newline c_{xy} & c_{yy} & c_{yz} \newline c_{xz} & c_{yz} & c_{zz} \end{bmatrix} $$
such that, $c_{ij} = c_{ji} = \frac{\partial^2c}{\partial i \partial j}$. The characteristic equation $\text{det} (\nabla^2 c - \lambda I) = 0$ of the Hessian then becomes,
$$ \begin{array}+ +\lambda^3 \newline -\lambda^2 (c_{xx} + c_{yy} + c_{zz}) \newline -\lambda\phantom{^1} (c_{xy}^2 + c_{xz}^2 + c_{yz}^2 - c_{xx}c_{yy} - c_{xx}c_{zz} - c_{yy}c_{zz}) \newline +\phantom{\lambda^0} (c_{xy}^2 c_{zz} + c_{xz}^2 c_{yy} + c_{yz}^2 c_{xx} - c_{xx}c_{yy}c_{zz} - 2c_{xy}c_{xz}c_{yz}) = 0 \end{array} $$
Given that the Hessian is symmetric such that the eigenvalues are real, what is the general solution of the characteristic equation assuming that it exists?
Are the eigenvalues of this Hessian at a critical point $\nabla c = 0$ the principal curvatures at that location?